A frequent objective in quality assurance in numerous industrial processes, or also in research and medical diagnostics, is the quantitative examination of the composition of a collected sample, in particular the determination of the concentration of a certain substance in the sample. In order to determine the concentration of a substance in a sample, the substance contained in the sample can be quantitatively converted with a reagent, for example until a color change of an added indicator indicates the end of the conversion reaction, while the quantity of the added reagent is tracked. However, such conventional chemical methods are quite laborious and have recently been replaced in large part by spectroscopic methods. With spectroscopic methods, the interaction of the substance in the sample with an investigatory radiation is used to determine the concentration. Such a spectroscopic method is disclosed for example in DE 10 2014 203 721 A1 (=Reference [2])
A powerful spectroscopic method of quantity analytical chemistry is nuclear magnetic resonance (NMR) spectroscopy. Here, typically, in one-dimensional NMR spectroscopy, the nuclear spins in the sample are aligned in a strong, static magnetic field, and the nuclear magnetization is rotated through 90° with a high-frequency pulse. The high-frequency response of the sample is then recorded as a function of time (referred to as a free induction decay (FID) signal). A frequency spectrum of the sample containing characteristic peaks for the individual constituents of the sample can be obtained from the time signal by Fourier transformation, wherein the individual peaks of the constituents overlap to a greater or lesser extent.
The intensity of the peaks of the individual constituents of the sample is basically proportional to the concentration of the associated constituent in the sample. However, due to the overlapping of a multiplicity of peaks in a spectrum, it is often not easy to quantitatively determine the signal component belonging to a particular substance. This not only applies to NMR, but also to other spectroscopic methods such as infrared (IR) spectroscopy or x-ray spectroscopy (x-ray fluorescence or x-ray absorption).
It is also known to identify the individual peaks associated with a substance and to determine the relative positions, intensities, line widths and line forms in a reference spectrum of the substance to be quantified. The peaks can then be fitted to the measured spectrum of the sample and, in turn, integration under the fitted peaks can be carried out in order to quantify the signal content of the substance. This so-called multiplet approach is implemented, for example, in the “Chenomx NMR Suite” spectral analysis software from Chenomx Inc., Edmonton, Alberta, Calif.
The limit of quantification and the error in the quantification in the determined concentration of a substance to be investigated are very important for assessing the results when evaluating spectra.
The limit of quantification (LOQ) is the lowest concentration of a substance in a measuring sample which can be quantitatively determined with a defined precision, e.g. with the relative error. Measuring results are usually only declared when this value is reached.
The limit of detection (LOD) is the lowest concentration of a substance in a measuring sample at which the substance can still be reliably detected. For example, the limit of detection can be determined by choosing the concentration above which the determined concentration becomes >0. All values below this concentration are designated as “undetectable.”
The integral of the signal of a substance is generally directly proportional to the concentration. In optical spectroscopy (IR, ultraviolet (UV), visual), this fact is described by the Beer-Lambert law; the behavior is similar in NMR spectroscopy.
Due to the linearity of signal and concentration, it is possible to produce artificial mixtures by adding spectra (see, for example, Reference [3]). The addition of a defined quantity of a substance to a mixture is referred to as “spiking.” If instead of the actual addition of a substance, a spectrum of the substance with a defined integral is added to a spectrum of a mixture, this is referred to as “electronic spiking.”
A substance in a spectrum can be quantified using different known methods, such as:                1. Integration of a region by summing the intensity values.        2. Identification of the individual signals with subsequent summation of the individual integrals. and        3. Adaptation of a line form (“model”) to an experimental signal (see Reference [2]).        
The above methods 2 and 3 use iterative algorithms to adapt the individual parameters, e.g., Levenberg-Marquardt or Gauss-Newton. Internally, the least squares method is normally used in each iteration step to calculate the deviation between model and experiment. This method has proved useful in practice. In addition, for practical reasons (above all computation time), a tolerance is assumed. If the values change during the iteration by less than the specified tolerance, the calculation is terminated. For example, the iteration is terminated if the change lies within the tolerance during the calculation of the target accuracy.
Quantifications according to methods 2 and 3 provide more accurate results but require an appreciable amount of computation time. It is therefore important to limit the number of steps necessary in a quantification as much as possible.
The accuracy of the quantification result depends on many factors. One of these factors is the spectral background which, among other things, includes:                Noise.        Other signals in the region which overlap with the substance.        
These effects play an ever decreasing role with increasing substance concentration, since the relative error becomes smaller.
Usual methods for determining LOQ, LOD (according to Reference [1]) include:                Visual definition.        Signal/noise ratio: LOD: 2-3*noise level, LOQ: 10*noise level.        Standard deviation in a spectrum of a blank sample according to Equation 1.        Calculation based on the calibration line (e.g. linear regression) at low concentration according to the following equation:LOD/LOQ=F*SD/b, where  (Equation 1)                    F: Factor, e.g. for LOD, F=3.3; for LOQ, F=10.0. Accepted values in practice [1].            SD: Standard deviation. Examples according to Reference [1] include:                        Values in the noise region in the blank spectrum.        Residues of linear regression of the calibration line.        Slope of the calibration line/Values in the spectrum.        
The present invention builds on the prior art according to a technique used in the Bruker “AssureNMR” software (see Reference [4]), which uses the following steps to determine LOQ:                1. First, a quantification is carried out at concentration 0, that is to say with the pure, “blank” spectrum. If the substance is not identified, the relative error in this case is set to “infinity,” as the expected concentration is zero.        2. A start concentration is determined. For practical reasons, the concentration “1” is assumed regardless of the units. The sample spectrum is then calculated and subsequently quantified. If the quantification is not successful, the concentration is multiplied by a factor of 10 until a concentration >0 and an error less than the aimed-for relative error are obtained.        3. Iteration using the binary search method:                    a. Determination of the current concentration=Mean value of greatest concentration>max. relative error and lowest concentration<max. relative error.            b. Electronic spiking+quantification.            c. Determination of the relative error.            d. Distance between greatest concentration>max. relative error and lowest concentration<max. relative error.            e. Is this distance less than the required accuracy? Then terminate.                        4. LOQ is the lowest concentration <max. relative error within the allowed tolerance.        
The number of iterations can become very high; to a certain extent the search is carried out blind. The method requires a large number of iteration steps and is relatively slow due to the many quantification steps to be carried out.
In addition, heretofore, there has been no way of estimating the accuracy of quantification results for the currently determined concentration. The accuracy as a function of the concentration has not previously been estimated.
Known methods and conventional solutions therefore have at least the following disadvantages or shortcomings:                It is impossible or nearly impossible to determine the error for a given concentration.        Concentration series (electronic and mechanical) must be set manually.        Overly simple formula: Multiplying the signal/noise ratio by a single factor does not take into account the chemical background: Signals in the region can, however, be due to other substances, possibly in low concentration.        Determination of LOQ at different specified accuracy values EB with simply a one-off calculation of the model function. In practice, two limits of quantification are frequently specified, for example 1% and 5%.        